An Introduction to G-Convergence
The last twentyfive years have seen an increasing interest for variational convergences and for their applications to different fields, like homogenization theory, phase transitions, singular perturbations, boundary value problems in wildly perturbed domains, approximation of variatonal problems, and non smooth analysis. Among variational convergences, De Giorgi's r-convergence plays a cen tral role for its compactness properties and for the large number of results concerning r -limits of integral functionals. Moreover, almost all other varia tional convergences can be easily expressed in the language of r -convergence. This text originates from the notes of the courses on r -convergence held by the author in Trieste at the International School for Advanced Studies (S. I. S. S. A. ) during the academic years 1983-84,1986-87, 1990-91, and in Rome at the Istituto Nazionale di Alta Matematica (I. N. D. A. M. ) during the spring of 1987. This text is far from being a treatise on r -convergence and its appli cations.
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Anal appear Appl applications Assume Banach space boundary bounded Chapter closed coercive compact consider constant continuous converges Corollary corresponding countable definition denotes dense domains elliptic equations equi-coercive equivalent estimate Example exists fact finite functional defined functional F given hence holds homogenization implies increasing functional inequality inf Fn inf Fn(y inner regular integral functionals K-lim Lemma Let F lim inf inf lim sup limit linear lower semicontinuous LP(N Maso Math measure method minimizer minimum point Moreover neighbourhood non-negative norm Note obtain open subset operator optimal otherwise Paris particular periodic positive problems Proof properties Proposition prove quadratic form r-converges to F r-limits refer relaxation Remark resp respect satisfies sc-F sequence sequence Fn sequentially shows strong topology subsequence sup inf Suppose T-lim inf T-lim sup taking Theorem theory values variational weak topology weakly
Page 330 - An approach for constructing families of homogenized equations for periodic media. I: An integral representation and its consequences.
Page 327 - Un teorema di passaggio al limite per la somma di funzioni convesse, Boll.
Page 330 - Cont re-examples pour divers problemes ou le controle intervient dans les coefficients.
Page 332 - On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles.